The weighted Yamabe problem with boundary

TL;DR

This paper introduces a new weighted Yamabe flow on smooth metric measure spaces with boundary, proving its long-term existence and convergence, extending geometric analysis techniques to weighted settings.

Contribution

It develops a novel Yamabe-type flow for weighted scalar curvature on manifolds with boundary, establishing its long-time existence and convergence results.

Findings

Flow exists for all time

Flow converges to a metric with constant weighted scalar curvature

Provides a new approach for geometric analysis on weighted manifolds

Abstract

We introduce a Yamabe-type flow \begin{align*} \left\{ \begin{array}{ll} \frac{\partial g}{\partial t} &=(r^m_{\phi}-R^m_{\phi})g \\ \frac{\partial \phi}{\partial t} &=\frac{m}{2}(R^m_{\phi}-r^m_{\phi}) \end{array} \right. ~~\mbox{ in }M ~~\mbox{ and }~~ H^m_{\phi}=0 ~~\mbox{ on }\partial M \end{align*} on a smooth metric measure space with boundary $(M,g, v^mdV_g,v^mdA_g,m)$, where $R^m_{\phi}$ is the associated weighted scalar curvature, $r^m_{\phi}$ is the average of the weighted scalar curvature, and $H^m_{\phi}$ is the weighted mean curvature. We prove the long-time existence and convergence of this flow.